A nonlinear two compartmental fractional derivative model

Abstract

This study presents a new nonlinear two compartmental model and its application to the evaluation of valproic acid (VPA) pharmacokinetics in human volunteers after oral administration. We have used literature VPA concentrations. In the model, the integer order derivatives are replaced by derivatives of real order often called fractional order derivatives. Physically that means that the history (memory) of a biological process, realized as a transfer from one compartment to another, is taken into account with the mass balance conservation observed. Our contribution is the analysis of a specific nonlinear two compartmental model with the application in evaluation of VPA pharmacokinetics. The agreement of the values predicted by the proposed model with the values obtained through experiments is shown to be good. Thus, pharmacokinetics of VPA after oral application can be described well by a nonlinear two compartmental model with fractional derivatives of the same order proposed here. Parameters in the model are determined by the least-squares method and the particle swarm optimization (PSO) numerical procedure is used. The results show that the nonlinear fractional order two compartmental model for VPA pharmacokinetics is superior in comparison to the classical (integer order) linear two compartmental model and to the linear fractional order two compartmental model.

Appendix

In the sequel, we present a formal result regarding existence and uniqueness for the equations of the proposed model. The results are basis for the iterative procedure that can be used to construct the solution to a nonlinear fractional initial value problem (11), (12). We also propose a numerical procedure for evaluation of fractional order integrals and give an error estimate for such a procedure. The procedure is finally used to reduce a linear fractional differential equation to a system of linear algebraic equations, which can be solved in a standard way.

1.1 Reduction to integral equation

The Eq. 11 may be reduced to a Volterra integral equation. We state this as:

Theorem 1.

Let α ∈ (0, 1) and let F:[0, T] × D, D ⊆ ℜ, be a continuous bounded function, satisfying the Lipschitz condition in its second argument, i.e. let there exists a finite positive constant L such that for any t ∈ [0, T] and any x, y ∈ D

$$ |F(t,x) - F(t,y)| \le L|x - y|, $$

(21)

Then there exists a unique continuous solution to the initial value problem (11)

$$ {}_{0}D_{t}^{\alpha } q = F(t,q),\quad q(0) = q_{0} \in \Re . $$

(22)

In fact, the problem is equivalent to a Volterra integral equation

$$ q(t) = q_{0} + {}_{0}I_{t}^{\alpha } F(t,q(t)). $$

(23)

The solution of (23) is the limit of the sequence of continuous functions q ^k (t), defined as

$$ q^{k + 1} (t) = q_{0} + {}_{0}I_{t}^{\alpha } F(t,q^{k} (t)), $$

(24)

where

$$ q^{0} (t) = q_{0} . $$

(25)

Proof

See (Kilbas and Marzan 2005).

Note.

According to assumptions (2.A) and (4.A), the fractional order models addressed in the present study satisfy the conditions of Theorem 1.

Let us explore the successive approximations method further. It is shown in Podlubny (1999) that

$$ \mathop {\sup }\limits_{{\tau \in [0,t_{1} ]}} |q_{2} (t) - q_{1} (t)| \le \frac{{t_{1}^{\alpha } }}{\alpha \Upgamma (\alpha )}L\mathop {\sup }\limits_{{\tau \in [0,t_{1} ]}} |q_{1} (t) - q_{0} (t)|. $$

(26)

One can always choose t ₁ ∈ (0, T] so that

$$ \frac{{t_{1}^{\alpha } }}{\alpha \,\Upgamma (\alpha )}L < 1, $$

(27)

and thus, according to the Banach fixed point theorem, the unique solution to the initial value problem is $q(t) = \mathop {\lim }\nolimits_{k \to \infty } q^{k} (t).$

The solution constructed in such a way is defined only for t ∈ [0, t ₁]. However, this solution can be extended on the interval t ∈ [t ₁, 2t ₁]. For such a t, (23) becomes

$$ \begin{aligned} q(t) & = q_{0} + {}_{0}I_{t}^{\alpha } F(t,q(t)) = q_{0} + \frac{1}{\Upgamma (\alpha - 1)}\int\limits_{0}^{{t_{1} }} {F(t,q(t))(t - \tau )^{\alpha - 1} d\tau } \\ & = q_{0} + \frac{1}{\Upgamma (\alpha - 1)}\int\limits_{{t_{1} }}^{t} {F(t,q(t))(t - \tau )^{\alpha - 1} d\tau } \\ \\ & \quad+ \frac{1}{\Upgamma (\alpha - 1)}\int\limits_{0}^{t} {F(t,q(t))(t - \tau )^{\alpha - 1} d\tau .} \\ \end{aligned} $$

(28)

Since the solution is know in the interval t ∈ [0, t ₁], the first two terms in the above expression can be considered as a known function of time. Denote this function by y. Then

$$ y(t) = q_{0} + \frac{1}{\Upgamma (\alpha - 1)}\int\limits_{{t_{1} }}^{t} {F(t,q(t))(t - \tau )^{\alpha - 1} d\tau } . $$

(29)

Integral equation (28) can be rewritten as

$$ q(t) = y(t) + \frac{1}{\Upgamma (\alpha - 1)}\int\limits_{0}^{{t_{1} }} {F(t,q(t))(t - \tau )^{\alpha - 1} d\tau } . $$

(30)

The above equation can be solved by successive approximation method, in the same way as Eq. 23, except that the domain of convergence is different. Following the same procedure repeatedly, one can expand the solution of the fractional initial value problem to the entire interval t ∈ [0, T].

1.2 Numerical procedure for evaluation of the fractional integral

We present now a numerical method to evaluate the fractional integral

$$ _{0} I_{t}^{\alpha } f = \int\limits_{0}^{t} {g\left( \tau \right)\left( {t - \tau } \right)^{\alpha - 1} d\tau } , $$

(31)

with g(t) an arbitrary function having bounded first derivative. This type of integral appears in (28). Let us denote the time interval under consideration into N equally sized subinterval each of length Δ (T = NΔ). Assume that f can be approximated by a piecewise constant, left-continuous signal, i.e., assume

$$ g(t) \approx g(k\Updelta ),\quad k\Updelta < t \le k\Updelta + \Updelta . $$

(32)

Left fractional integral of signal g evaluated at t = kΔ can then be approximated as

$$ _{0} I_{k\Updelta }^{\alpha } g = \frac{1}{\Upgamma (\alpha )}\int\limits_{0}^{k\Updelta } {g(k\Updelta - \tau )\tau^{\alpha - 1} d\tau } = \sum\limits_{i = 0}^{k - 1} {\frac{1}{\Upgamma (\alpha )}\int\limits_{i\Updelta }^{i\Updelta + \Updelta } {g(k\Updelta - \tau )\tau^{\alpha - 1} d\tau } \approx } \sum\limits_{i = 0}^{k - 1} {\frac{1}{\Upgamma (\alpha )}g(k\Updelta - i\Updelta )\int\limits_{i\Updelta }^{i\Updelta + \Updelta } {\tau^{\alpha - 1} d\tau } } . $$

Let us introduce

$$ w_{i}^{\alpha } = \frac{1}{\Upgamma (\alpha )}\int\limits_{i\Updelta }^{i\Updelta + \Updelta } {\tau^{\alpha - 1} d\tau } = \frac{{\Updelta^{\alpha } }}{\Upgamma (\alpha + 1)}\left[ {(i + 1)^{\alpha } - i^{\alpha } } \right]. $$

(33)

The approximation can now be seen as a discrete convolution of signals g _i  = g(iΔ) and w ^α _i :

$$ _{0} I_{k\Updelta }^{\alpha } g \approx \sum\limits_{i = 0}^{k - 1} {w_{i} g_{k - i} } . $$

(34)

The above formula represents a generalization of the forward rectangular rule used for numerical evaluation of classical, first order integrals. The formula is also known as the right rectangular or first Euler rule.

The approximation is meaningful only if g(iΔ) is defined for each i, i.e., if g is a bounded signal. The following theorem gives an upper bound on the approximation error in the case that g has bounded first derivative also.

Theorem 2.

Let N be an integer and Δ a real number. Let g:[0, NΔ] → ℜ be a continuously differentiable function such that

$$ M = \mathop {\sup }\limits_{0 \le \tau \le N\Updelta } |g^{(1)} (\tau )| < \infty . $$

(35)

The following bound holds for the absolute error e ^α _k of the approximation formula for fractional integral

$$ e_{k}^{\alpha } = \left| {_{0} I_{k\Updelta }^{\alpha } - \sum\limits_{i = 0}^{k - 1} {w_{i} g_{k - i} } } \right| \le \frac{{\Updelta^{\alpha + 1} }}{\Upgamma (\alpha + 2)}M\eta (\alpha ,k), $$

(36)

where

$$ \eta (\alpha ,k) = \sum\limits_{i = 0}^{k - 1} {\left| {(\alpha + 1)i(i + 1)^{\alpha } - i^{\alpha + 1} - \alpha (i + 1)^{\alpha + 1} } \right|} . $$

(37)

Proof

By virtue of Taylor expansion, for any τ ∈ (iΔ, iΔ + Δ) there exists a σ _τ  ∈ (iΔ, iΔ + Δ). such that

$$ g(k\Updelta - \tau ) = g(k\Updelta - i\Updelta ) - g^{(1)} (\sigma_{\tau } )(i\Updelta - \tau ), $$

Therefore,

$$ \begin{aligned}_{0} I_{k\Updelta }^{\alpha } g & = \frac{1}{\Upgamma (\alpha )}\int\limits_{0}^{k\Updelta } {g(k\Updelta - \tau )\tau^{\alpha - 1} d\tau } = \sum\limits_{i = 0}^{k - 1} {\frac{1}{\Upgamma (\alpha )}\int\limits_{i\Updelta }^{i\Updelta + \Updelta } {g(k\Updelta - \tau )\tau^{\alpha - 1} d\tau } } \\ & = \sum\limits_{i = 0}^{k - 1} {\frac{1}{\Upgamma (\alpha )}\int\limits_{i\Updelta }^{i\Updelta + \Updelta } {\left[ {g(k\Updelta - i\Updelta ) - g^{(1)} (\sigma_{\tau } )(i\Updelta - \tau )} \right]\tau^{\alpha - 1} d\tau } } \\ & = \sum\limits_{i = 0}^{k - 1} {\frac{1}{\Upgamma (\alpha )}g(k\Updelta - i\Updelta )\int\limits_{i\Updelta }^{i\Updelta + \Updelta } {\tau^{\alpha - 1} d\tau } } \\ & \quad - \sum\limits_{i = 0}^{k - 1} {\frac{1}{\Upgamma (\alpha )}\int\limits_{i\Updelta }^{i\Updelta + \Updelta } {g^{(1)} (\sigma_{\tau } )(i\Updelta - \tau )\tau^{\alpha - 1} d\tau } } . \\ \end{aligned} $$

The first sum in the above expression is, in fact, the approximation, while the second sum is the approximation error. Therefore, we have

$$ \begin{aligned} e_{k}^{\alpha } & = \left| {_{0} I_{k\Updelta }^{\alpha } g - \sum\limits_{i = 0}^{k - 1} {w_{i} g_{k - i} } } \right| = \left| {\sum\limits_{i = 0}^{k - 1} {\frac{1}{\Upgamma (\alpha )}\int\limits_{i\Updelta }^{i\Updelta + \Updelta } {g^{(1)} (\sigma_{\tau } )(i\Updelta - \tau )\tau^{\alpha - 1} d\tau } } } \right| \le \sum\limits_{i = 0}^{k - 1} {\frac{M}{\Upgamma (\alpha )}\left|\int\limits_{i\Updelta }^{i\Updelta + \Updelta } {(i\Updelta - \tau )\tau^{\alpha - 1} d\tau } \right|} \\ & = \sum\limits_{i = 0}^{k - 1} {\frac{{M\Updelta^{\alpha + 1} }}{\Upgamma (\alpha + 2)}} \left| {(\alpha + 1)i(i + 1)^{\alpha } - i^{\alpha + 1} - \alpha (i + 1)^{\alpha + 1} } \right|. \\ \end{aligned} $$

This concludes the proof.

1.3 The simulation procedure

The Theorem above states that fractional integral of a function evaluated at time t = kΔ can be approximated by a linear combination of the function values at times t = iΔ, where 0 ≤ i ≤ k. The coefficients of this linear combination are w ^α _i . In other words, if one denotes by g a vector with ith entry equal to g(iΔ),  and by I ^α g a vector with ith entry equal to the approximation of $_{0} I_{i\Updelta }^{\alpha } g$,  then

$$ I^{\alpha } {\mathbf{g}} = {\mathbf{I}}^{\alpha } \cdot {\mathbf{g}}, $$

(38)

where “·” denotes matrix–vector multiplication and I ^α is a matrix whose element at position (i, j) is $w_{i - j}^{\alpha }$ if i > j and 0 otherwise. Denote by q ₁, f(q ₁) and q ₂ vectors containing values of q ₁, f(q ₁) and q ₂, respectively. The method of successive approximations is implemented as

$$ {\mathbf{q}}_{1}^{k} = q_{1} (0){\mathbf{1}} - k_{21} {\mathbf{I}}^{\alpha } \cdot {\mathbf{q}}_{1}^{k + 1} ,\;k \ge 1, $$

(39)

with the initial approximation being

$$ {\mathbf{q}}_{1}^{0} = q_{1} (0){\mathbf{1}}, $$

(40)

where 1 is a vector with all entries equal to 1. In most cases considered in the current study, the difference between the two consecutive approximations dropped below 0.001 after less than 500 iterations.

The second equation is equivalent to the following

$$ q_{2} (t) = - k_{02} {}_{0}I_{t}^{\alpha } q_{2} + k_{21} {}_{0}I_{t}^{\alpha } f(q_{1} ), $$

(41)

and can, therefore, be approximated as

$$ {\mathbf{q}}_{2} = - k_{02} {\mathbf{I}}^{\alpha } \cdot {\mathbf{q}}_{2} + k_{21} {\mathbf{I}}^{\alpha } \cdot f({\mathbf{q}}_{1} ), $$

(42)

which can be solved explicitly as

$$ {\mathbf{q}}_{2} = ({\mathbf{E}} + k_{02} {\mathbf{I}}^{\alpha } )^{ - 1} \cdot k_{21} {\mathbf{I}}^{\alpha } \cdot f({\mathbf{q}}_{1} ), $$

(43)

with E being the identity matrix. The above expression can be seen as numerical implementation of the solution obtained via the Laplace transform method (see Eq. 19).